NAG Library Routine Document

G05ZTF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     NS – INTEGERInput

On entry: the number of sample points (grid points) to be generated in realisations of the increments of the fractional Brownian motion. This must be the same value as supplied to G05ZMF or G05ZNF when calculating the eigenvalues of the embedding matrix.

Constraint: $\mathbf{NS}\ge 1$.

2:     S – INTEGERInput

On entry: the number of realisations of the fractional Brownian motion to simulate.

Constraint: $\mathbf{S}\ge 1$.

3:     M – INTEGERInput

On entry: the size of the embedding matrix, as returned by G05ZMF or G05ZNF.

Constraint: $\mathbf{M}\ge \max \left(1,2\left(\mathbf{NS}-1\right)\right)$.

4:     XMAX – REAL (KIND=nag_wp)Input

On entry: the upper bound for the interval over which the fractional Brownian motion is to be simulated, as returned by G05ZMF or G05ZNF.

Constraint: $\mathbf{XMAX}>0.0$.

5:     H – REAL (KIND=nag_wp)Input

On entry: the Hurst parameter for the fractional Brownian motion. This must be the same value as supplied to G05ZMF or G05ZNF when calculating the eigenvalues of the embedding matrix.

Constraint: $0.0<\mathbf{H}<1.0$.

6:     LAM(M) – REAL (KIND=nag_wp) arrayInput

On entry: contains the square roots of the eigenvalues of the embedding matrix, as returned by G05ZNF.

Constraint: $\mathbf{LAM}\left(i\right)=0$, $i=1,2,\dots ,\mathbf{M}$.

7:     RHO – REAL (KIND=nag_wp)Input

On entry: indicates the scaling of the covariance matrix, as returned by G05ZMF or G05ZNF.

Constraint: $0.0<\mathbf{RHO}\le 1.0$.

8:     STATE($*$) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

9:     Z($\mathbf{NS}+1$,S) – REAL (KIND=nag_wp) arrayOutput

On exit: contains the realisations of the fractional Brownian motion. Each column of Z contains one realisation of the fractional Brownian motion, with $\mathbf{Z}\left(i,j\right)$, for $j=1,2,\dots ,\mathbf{S}$, corresponding to the gridpoint $\mathbf{XX}\left(i\right)$.

10:   XX($\mathbf{NS}+1$) – REAL (KIND=nag_wp) arrayOutput

On exit: the gridpoints at which values of the fractional Brownian motion are output. The first gridpoint is always zero, and the subsequent NS gridpoints represent the equispaced steps towards the last gridpoint, XMAX. Note that in G05ZMF and G05ZNF, the returned NS sample points are the mid-points of the grid returned in XX here.

11:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry, $\mathbf{NS}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{NS}\ge 1$.

$\mathbf{IFAIL}=2$

On entry, $\mathbf{S}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{S}\ge 1$.

$\mathbf{IFAIL}=3$

On entry, $\mathbf{M}=⟨\mathit{\text{value}}⟩$, and $\mathbf{NS}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{M}\ge \max \left(1,2\left(\mathbf{NS}-1\right)\right)$.

$\mathbf{IFAIL}=4$

On entry, $\mathbf{XMAX}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{XMAX}>0.0$.

$\mathbf{IFAIL}=5$

On entry, $\mathbf{H}=⟨\mathit{\text{value}}⟩$.
Constraint: $0.0<\mathbf{H}<1.0$.

$\mathbf{IFAIL}=6$

On entry, at least one element of LAM was negative.
Constraint: all elements of LAM must be non-negative.

$\mathbf{IFAIL}=7$

On entry, $\mathbf{RHO}=⟨\mathit{\text{value}}⟩$.
Constraint: $0.0<\mathbf{RHO}\le 1.0$.

$\mathbf{IFAIL}=8$

On entry, STATE vector has been corrupted or not initialized.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (g05ztfe.f90)

9.2  Program Data

Program Data (g05ztfe.d)

9.3  Program Results

Program Results (g05ztfe.r)